This post describes a new, powerful tool, called a dot-dot simulation, for experiencing function behavior when both the input and output variables of the function describe discrete units. To use the tool, you enter a function, enter a minimum and maximum value for the domain, as well as a minimum and maximum value for the range. The tool then allows you to visualize the input and output pairs of the function as side-by-side columns of stacked dots.
Here’s a detailed picture of how the tool works:
Here’s an image of the tool cycling through some inputs for the function 
And here is the tool itself. To acclimate yourself with the tool:
- Slide the slider from left to right and notice that it goes from -10 to 10.
- Change the minimum and maximum values for
to -6 and 9. Then slide the slider again and notice that it now goes from -6 to 9.
- Notice how the vertical axis goes from -10 to 10. Change the minimum and maximum values for
to -20 and 30. Notice how the vertical axis changes to reflect the new range.
- Finally, change the function to something other than
, then slide the slider again and note how the dot stacks reflect the new function’s values.
DOT DOT SIMULATION
Finally, here’s a task for you to try that illustrates how easily this tool breathes meaning into even the most obscure functions and situations.
Imagine a story about food pellets and fish. Let 
- Create a dot dot simulation for this situation and examine the output for 0 to 100 pellets.
- Make up a story that explains this data.
I gave my 7th graders 15 minutes to complete this same task. Here’s what some of them said.
Student 1:
Student 2:
Student 3:
Student 4:
Even though, some of the stories are crude, the exciting thing here is that 7th graders constructed creative, meaningful interpretations of the function